Eibar went into the contest unbeaten in three league meetings with the Galicians and have now kept four consecutive clean sheets against Oscar's side.
In an even game of few chances, Celta were unable to find a cutting edge even after Pape Diop received two bookings in the space of six minutes in the closing stages.
Celta have now failed to win any of their previous four opening-day games in the Basque County and have only come away with victory once in their past 11 trips to the region.
Eibar boss Jose Luis Mendilibar boasts an enviable record against them, having lost only once in eight home league games, keeping five clean sheets in the process.
The 2020-21 season continues on Saturday with Granada hosting Athletic Bilbao before newly promoted Cadiz take on Osasuna.
Reigning champions Real Madrid are not in action until September 20, when they visit Real Sociedad.